Quadratic Equations
Solve by factoring, completing the square or the quadratic formula; use the discriminant.
A quadratic equation can be written as with . Its graph is a parabola, and the solutions are the -intercepts. There can be two real solutions, one repeated solution, or none, depending on whether the parabola crosses, touches or misses the -axis.
Three methods solve every quadratic you will meet: factoring when the roots are tidy, completing the square, and the quadratic formula , which always works. The discriminant predicts the number of real solutions before you compute them.
Key terms
- Quadratic formula
- The roots of a quadratic in standard form.
- Discriminant
- Its sign tells whether there are two real, one repeated, or two complex roots.
Key ideas
- Set the equation equal to zero before factoring; the zero-product property needs a zero on one side.
- Discriminant positive: two real roots; zero: one repeated root; negative: two complex roots.
- Sum of roots is and product is , a quick check.
- The vertex lies at , halfway between the roots.
Worked example
Solve .
- Discriminant: .
- Formula: .
- Roots: and .
- Check the sum: .
Answer: or
Common mistakes
- Dividing both sides by and losing the root .
- Forgetting the and reporting only one root.
- Dividing only by instead of the whole numerator.
Practice questions
Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.
1.What is the larger root of ?
Show answer
2.What is the discriminant of ?
Show answer
3.What is the positive root of ?
Show answer